Propagation through and characterization of atmospheric and oceanic phenomena: introduction to the joint feature issue in Applied Optics and Journal of the Optical Society of America A.

This joint feature issue in Applied Optics and JOSA A collects articles focused on the topic of propagation through and characterization of atmospheric oceanic phenomena. The papers cover a broad range of topics, many of which were addressed at the 2023 Propagation Through and Characterization of Atmospheric Oceanic Phenomena (pcAOP) Topical Meeting at the Optica Imaging Congress in Boston, Massachusetts, 14-17 August 2023. These papers are supplemented by numerous examples of the current state of research in the field. This is the first pcAOP feature issue, with the intention to produce an issue on this topic every two years.

Propagation through and characterization of atmospheric and oceanic phenomena: introduction to the joint feature issue in Applied Optics and Journal of the Optical Society of America A.

This joint feature issue in Applied Optics and JOSA A collects articles focused on the topic of propagation through and characterization of atmospheric oceanic phenomena. The papers cover a broad range of topics, many of which were addressed at the 2023 Propagation Through and Characterization of Atmospheric Oceanic Phenomena (pcAOP) Topical Meeting at the Optica Imaging Congress in Boston, Massachusetts, 14-17 August 2023. These papers are supplemented by numerous examples of the current state of research in the field. This is the first pcAOP feature issue, with the intention to produce an issue on this topic every two years.

Low-wavenumber compensation with Zernike tilt for non-Kolmogorov turbulence phase screens.

The fast-Fourier-transform-based filtering method for phase screen generation remains popular for numerical simulation of optical propagation through turbulence; however, these screens inherently underrepresent the spectral density at low wavenumbers. Here, the "Z-tilt" approach is explored to augment the spectral density at low wavenumbers by adding a random phase tilt, which is derived from the wavefront phase statistics of a Zernike polynomial basis. This approach is computationally efficient and can be applied to any statistically homogeneous and isotropic refractive index field. An analytic result is provided for the von Kármán spectrum with finite outer scale. In a quantitative comparison with phase screens compensated for using a common subharmonic approach, the Z-tilt method shows the best agreement with the analytical structure function when the outer scale is greater than about three times the screen dimension. For outer scales of the order of the screen dimension, the subharmonic and a modified Z-tilt method give the most accurate results. A propagation simulation demonstrates that the aperture-averaged angle-of-arrival variance is accurately predicted using the Z-tilt method.

Phase-factor spectra of turbulent phase screens.

The optical phase ϕ is a key quantity in the physics of light propagating through a turbulent medium. In certain respects, however, the statistics of the phase factor, ψ=exp⁡(iϕ), are more relevant than the statistics of the phase itself. Here, we present a theoretical analysis of the 2D phase-factor spectrum Fψ(κ) of a random phase screen. We apply the theory to four types of phase screens, each characterized by a power-law phase structure function, Dϕ(r)=(r/rc)γ (where rc is the phase coherence length defined by Dϕ(rc)=1rad2), and a probability density function pα(α) of the phase increments for a given spatial lag. We analyze phase screens with turbulent (γ=5/3) and quadratic (γ=2) phase structure functions and with normally distributed (i.e., Gaussian) versus Laplacian phase increments. We find that there is a pronounced bump in each of the four phase-factor spectra Fψ(κ). The precise location and shape of the bump are different for the four phase-screen types, but in each case it occurs at κ∼1/rc. The bump is unrelated to the well-known "Hill bump" and is not caused by diffraction effects. It is solely a characteristic of the refractive-index statistics represented by the respective phase screen. We show that the second-order ψ statistics (covariance function, structure function, and spectrum) characterize a random phase screen more completely than the second-order ϕ counterparts.

Optical timing jitter due to atmospheric turbulence: comparison of frequency comb measurements to predictions from micrometeorological sensors.

During propagation through atmospheric turbulence, variations in the refractive index of air cause fluctuations in the time-of-flight of laser light. These timing jitter fluctuations are a major noise source for precision laser ranging, optical time transfer, and long-baseline interferometry. While there exist models that estimate the turbulence-induced timing jitter power spectra using parameters obtainable from conventional micrometeorological instruments, a direct and independent comparison of these models to measured timing jitter data has not been done. Here we perform this comparison, measuring turbulence-induced optical pulse timing jitter over a horizontal, near-ground path using frequency comb lasers while independently characterizing the turbulence along the path using a suite of micrometeorological sensors. We compare the power spectra of measured optical pulse timing jitter to predictions based on the measured micrometeorological data and standard turbulence theory. To further quantitatively compare the frequency comb data to the micrometeorological measurements, we extract and compare the refractive index structure parameter, Cn2, from both systems and find agreement to within a factor of 5 for wind speed >1 m/s, and further improvement is possible as wind speed increases. These results validate the use of conventional micrometeorological instruments in predicting optical timing jitter statistics over co-located laser beam paths.

Optical propagation through non-overturning, undulating temperature sheets in the atmosphere.

It is a standard assumption in the theory of optical propagation through the turbulent atmosphere that the refractive-index fluctuations n1(x) are statistically isotropic. It is well known, however, that n1(x) in the free atmosphere and in the nocturnal boundary layer is often strongly anisotropic, even at very small scales. Here we present and discuss a model atmosphere characterized by randomly undulating, non-turbulent and non-overturning, quasi-horizontal refractive-index interfaces, or "sheets." We assume n1(x)=v[z-h(x,y)], where v(z) is a random function that has a 1D spectrum V(κz), and where h(x,y) is a vertical displacement that varies randomly as a function of the horizontal coordinates x and y. We derive a closed-form expression for the 3D spectrum Φ(κ) and show that the horizontal 1D spectra have the same power law as V(κz) if the structure function of h(x,y) is quadratic. Moreover, we evaluate the scintillation index σI2 for a plane wave propagating horizontally through the undulating sheets, and we compare σI2 predicted for undulating sheets with Tatarskii's classical predictions of σI2 for fully developed, isotropic turbulence. For Phillips-type sheets, where V(κz)∝κz-2, in the diffraction limit we find σI2∝k (where k=2π/λ is the optical wavenumber), which is only slightly different from Tatarskii's famous k7/6 law for propagation through fully developed, Obukhov-Corrsin-type, isotropic turbulence where Φ(κ)∝κ-11/3. Our model predicts that σI2 is inversely proportional to the sheet tilt angle standard deviation 〈θx2〉, regardless of whether or not diffraction plays a role and regardless of the value of the power-law exponent of V(κz).

Temperature variance dissipation equation and its relevance for optical turbulence modeling.

The 3D spectrum Φ(κ) of the turbulent air temperature fluctuations is a key quantity for the physics of optical propagation through the turbulent atmosphere. The standard model, which was derived in the 1950s by Tatarskii from the Obukhov-Corrsin theory of homogeneous and isotropic turbulence, is Φ(κ)=0.033CT2κ(-11/3)h(κl(0)), where κ=|κ| is the wavenumber, CT2 is the temperature structure parameter, l(0) is the inner temperature scale, and h(κl(0) is a universal function that approaches 1 for wavenumbers in the inertial range and drops to zero for κl(0)≫1. Certain performance characteristics of optical systems, such as the scintillation index for small receiving apertures, depend sensitively on the functional form of h(y) at y≈1. During the last 70 years, the optical-turbulence community has developed and applied various heuristic h(y) models. There is a constraint that any valid h(y) model has to fulfill: ∫0∞h(y)y(1/3)dy=(27/10)Γ(1/3)=7.233. This constraint is a dimensionless form of the spectral temperature variance dissipation equation, which follows directly from first-principle fluid mechanics. We show that Tatarskii's cutoff (1961) and Gaussian (1971) models fulfill this constraint, while three more recent models, including the widely used Andrews model [J. Mod. Opt.39, 1849 (1992)JMOPEW0950-034010.1080/09500349214551931], do not. The dissipation constraint can be used to "recalibrate" the coefficients in these models.

Investigation of Hill's optical turbulence model by means of direct numerical simulation.

For almost four decades, Hill's "Model 4" [J. Fluid Mech. 88, 541 (1978) has played a central role in research and technology of optical turbulence. Based on Batchelor's generalized Obukhov-Corrsin theory of scalar turbulence, Hill's model predicts the dimensionless function h(κl(0), Pr) that appears in Tatarskii's well-known equation for the 3D refractive-index spectrum in the case of homogeneous and isotropic turbulence, Φn(κ)=0.033C2(n)κ(-11/3) h(κl(0), Pr). Here we investigate Hill's model by comparing numerical solutions of Hill's differential equation with scalar spectra estimated from direct numerical simulation (DNS) output data. Our DNS solves the Navier-Stokes equation for the 3D velocity field and the transport equation for the scalar field on a numerical grid containing 4096(3) grid points. Two independent DNS runs are analyzed: one with the Prandtl number Pr=0.7 and a second run with Pr=1.0 . We find very good agreement between h(κl(0), Pr) estimated from the DNS output data and h(κl(0), Pr) predicted by the Hill model. We find that the height of the Hill bump is 1.79 Pr(1/3), implying that there is no bump if Pr<0.17 . Both the DNS and the Hill model predict that the viscous-diffusive "tail" of h(κl(0), Pr) is exponential, not Gaussian.

Effects of wind-driven telescope vibrations on measurements of turbulent angle-of-arrival fluctuations.

Turbulence in the atmospheric refractive-index field causes optical angle-of-arrival (AOA) fluctuations that can be used for atmospheric remote sensing of various parameters, including wind velocities and the optical refractive-index turbulence structure parameter, C(n)2. If AOA measurements are contaminated by wind-induced telescope vibrations, the underlying retrieval algorithms may fail. In order to study the effects of wind-driven telescope vibrations on optical-turbulence measurements, we conducted a field experiment in which we exposed two small telescopes deliberately to the wind. We measured AOA fluctuations of visible light propagating along a horizontal, 174 m long path 1.7 m above flat terrain, and we used fast-response ultrasonic anemometers to measure the wind velocity at multiple locations along the path. We found (1) that the AOA turbulence spectra were contaminated by multiple resonance peaks, (2) that the resonance frequencies were independent of the wind speed, and (3) that the AOA variance associated with the dominating vibration mode was proportional to the fourth power of the wind speed.

Optical anemometry based on the temporal cross-correlation of angle-of-arrival fluctuations obtained from spatially separated light sources.

The temporal cross-correlation function of the angle-of-arrival (AOA) fluctuations of two optical waves propagating through atmospheric turbulence carries information regarding the average wind velocity transverse to the propagation path. We present and discuss two estimators for the retrieval of the path-averaged beam-transverse horizontal wind velocity, v(t). Both methods retrieve v(t) from the temporal cross-correlation function of AOA fluctuations obtained from two closely spaced light-emitting diodes (LEDs). The first method relies on the time delay of the peak (TDP) of the cross-correlation function, and the second method exploits its slope at zero lag (SZL). Over a 9 h period during which v(t) varied between -1.3 ms(-1) and 2.0 ms(-1), the maximum rms difference between optically retrieved and in situ measured 10 s estimates of v(t) was found to be 0.18 ms(-1) for the TDP estimator and 0.23 ms(-1) for the SZL estimator. Applicability and limitations of these two optical wind retrieval techniques are discussed.

Angle-of-arrival anemometry by means of a large-aperture Schmidt-Cassegrain telescope equipped with a CCD camera.

The frequency spectrum of angle-of-arrival (AOA) fluctuations of optical waves propagating through atmospheric turbulence carries information of wind speed transverse to the propagation path. We present the retrievals of the transverse wind speed, upsilon b, from the AOA spectra measured with a Schmidt-Cassegrain telescope equipped with a CCD camera by estimating the "knee frequency," the intersection of two power laws of the AOA spectrum. The rms difference between 30 s estimates of upsilon b retrieved from the measured AOA spectra and 30s averages of the transverse horizontal wind speed measured with an ultrasonic anemometer was 11 cm s(-1) for a 1 h period, during which the transverse horizontal wind speed varied between 0 and 80 cm s(-1). Potential and limitations of angle-of-arrival anemometry are discussed.

Closed-form approximations for the angle-of-arrival variance of plane and spherical waves propagating through homogeneous and isotropic turbulence.

The calculation of the aperture-averaged angle-of-arrival variance, observed with a telescope with a circular aperture, of a plane or spherical wave propagating through homogeneous and isotropic turbulence is one of the classical problems in the theory of wave propagation through random media. We present and discuss approximate closed-form solutions on the basis of the Rytov approximation. For both plane and spherical waves, the accuracy of the approximations is better than 0.25% for all ratios of aperture diameter and Fresnel length.